In the following, a network is discussed that consists of several multi-physical elements. The network elements describing the electric contribution are provided by current sources, voltage sources, nodes, ground, resistors, capacitors, and inductors. The fluid network consists of pipes, pumps, demands, junctions, and reservoirs. The electro-thermal coupling is established by lumped mass elements representing the pipe wall and the masses from the battery and heat transfer connections. The individual components are assembled to a network \(\mathcal{N}\), which is represented by a linear directed graph. The graph structure is described by an incidence matrix *A*, which can be used for the model descriptions, cf. [9]. In the following we state the DAEs for the three main involved physical networks.

### Electrical network

The electrical network \(\mathcal{N}_{E} = \lbrace R, C, L, V, I, N, G, B \rbrace \) is composed of resistors *R*, capacitors *C*, inductors *L*, voltage sources *V*, current sources *I*, nodes *N*, grounds *G* and batteries *B*. The DAE for the network in \(\mathcal{N}_{E}\) in input-output form is given by: For predefined continuous inputs \(u=(u_{R}^{T}, u_{C}^{T}, u_{B}^{T})^{T}\) find the potentials \(e = (e_{N}^{T}, e_{G}^{T})^{T}\), the currents \(j = (j_{R}^{T}, j_{C}^{T}, j_{L}^{T}, j_{V}^{T}, j_{B}^{T})^{T}\) and the outputs \(y = y_{R}\), such that

$$ \begin{aligned} &A_{R} j_{R} + A_{C} j_{C} + A_{L} j_{L} + A_{V} j_{V} + A_{B} j_{B} + A_{I} \bar{j}_{I}= 0, \\ &r(u_{R}) j_{R} - A_{R}^{T} e= 0, \\ &j_{C} - \frac{d(c(u_{C}) A_{C}^{T} e)}{d t}= 0, \\ &l \frac{d j_{L}}{d t} - A^{T}_{L} e= 0, \\ &A^{T}_{V} e= \bar{v}_{V}, \\ &A^{T}_{B} e= \bar{v}_{B}(j_{B}, u_{B}), \\ &y_{R}= \bigl\vert j_{R}\circ A^{T}_{R} e \bigr\vert \end{aligned} $$

(1)

for given boundary conditions \(e_{G} = 0\) and predefined resistance *r*, capacitance *c* and inductance *l* as well as prescribed currents \(\bar{j}_{I}\) and prescribed voltages \(\bar{v}_{V}\) and \(\bar{v}_{B}\). The coupling variables are expressed as temperature of the resistor \(u_{R}\), the capacitor \(u_{C}\) and the battery \(u_{B}\) as well as the energy flux of the resistor \(y_{R}\), where ∘ denotes the Hadamard product, i.e. the elementwise vector product, and \(|\cdot |\) the elementwise absolute value.

### Solid network

We consider a solid network \(\mathcal{N}_{S} = \lbrace SW, LW, HT, HS, TB \rbrace \) that includes solid walls *SW*, lumped walls *LW*, heat transfers *HT*, heat sources *HS* and temperature boundaries *TB*. The DAE for the network \(\mathcal{N}_{S}\) in input-output form is specified by: For predefined continuous inputs \((u_{Hs_{S}}^{T},u_{Tb_{S}}^{T})^{T}\), find the temperatures \((T_{Sw}^{T}, T_{Lw}^{T})^{T}\), the heat fluxes \(H_{Ht_{S}}\) and the outputs \((y_{Sw}^{T}, y_{Lw}^{T}, y_{Ht_{S}}^{T})^{T}\), such that

$$ \begin{aligned} &m_{Sw} c_{p,Sw} \frac{d T_{Sw}}{d t}= A_{Sw,Ht_{S}} H_{Ht_{S}} + A_{Sw,Hs} H_{Hs} + A_{Sw,Hs_{u}}u_{Hs_{S}}, \\ &0= A_{Lw,Ht_{S}} H_{Ht_{S}} + A_{Lw,Hs} H_{Hs} + A_{Lw,Hs_{u}}u_{Hs_{S}}, \\ &H_{Ht_{S}}= c_{Ht_{S}} \bigl(A_{Sw,Ht_{S}}^{T} T_{Sw} + A_{Lw,Ht_{S}}^{T} T_{Lw} + A_{Tb,Ht_{S}}^{T} T_{Tb} + A_{Tb_{u},Ht_{S}}^{T}u_{Tb_{S}} \bigr), \\ &y_{Sw}= \bigl\vert \bigl(A_{Sw,Hs_{u}}^{T} + A_{Tb_{u},Ht_{S}} A_{Sw,Ht_{S}}^{T}\bigr) \bigr\vert T_{Sw}, \\ &y_{Lw}= \bigl\vert \bigl(A_{Lw,Hs_{u}}^{T} + A_{Tb_{u},Ht_{S}} A_{Lw,Ht_{S}}^{T}\bigr) \bigr\vert T_{Lw}, \\ &y_{Ht_{S}}= A_{Tb_{u},Ht_{S}} H_{Ht_{S}} \end{aligned} $$

(2)

for given boundary conditions \(H_{Hs} = \bar{H}_{Hs}\) and \(T_{Tb} = \bar{T}_{Tb}\) and predefined positive definite coefficient matrices \(m_{Sw}\), \(c_{p,Sw}\) and \(c_{Ht_{S}}\). The coupling variables are expressed as the energy fluxes \(u_{Hs_{S}}\) and \(u_{Tb_{S}}\) and the temperatures \(y_{Sw}\), \(y_{Lw}\) and \(y_{Ht_{S}}\).

### Fluid network

The fluid network \(\mathcal{N}_{F} = \lbrace PI, PU, DE, VJ, LJ, RE, HT, TB \rbrace \) consists of pipes *PI*, pumps *PU*, demands *DE*, volume junctions *VJ*, lumped junctions *LJ*, reservoirs *RE*, heat transfers *HT* and temperature boundaries *TB*. The DAE for the network \(\mathcal{N}_{F}\) in input-output form is given by: For predefined continuous inputs \((u_{Hs_{F}}^{T},u_{Tb_{F}}^{T})^{T}\), find the pressures \((p_{Lj}^{T}, p_{Vj}^{T})^{T}\), the mass flows \((q_{Pi}^{T}, q_{Pu}^{T})^{T}\), the temperatures \((T_{Vj}^{T}, T_{Lj}^{T})^{T}\), the heat fluxes \((H_{Ht_{F}}^{T}, H_{Pu}^{T}, H_{Pi}^{T})^{T}\) and the outputs \((y_{Vj}^{T}, y_{Lj}^{T}, y_{Ht_{F}}^{T})^{T}\), such that

$$\begin{aligned}& \frac{d q_{Pi}}{dt} = c_{1,Pi} \bigl(A_{Jc,Pi}^{T} p_{Jc} + A_{Re,Pi}^{T} p_{Re} \bigr) + c_{2,Pi} \textrm{diag} \bigl( \vert q_{Pi} \vert \bigr)q_{Pi} + c_{3,Pi}, \\& f_{Pu}(q_{Pu}) = A_{Jc,Pu}^{T} p_{Jc} + A_{Re,Pu}^{T} p_{Re}, \\& 0 = A_{Jc,Pi} q_{Pi} + A_{Jc,Pu} q_{Pu} + A_{Jc,De} q_{De}, \\& m_{Vj} c_{p, Vj}\frac{d T_{Vj}}{d t} = A_{Vj,Pi} H_{Pi} + A_{Vj,Pu} H_{Pu} \\& \hphantom{m_{Vj} c_{p, Vj}\frac{d T_{Vj}}{d t} =}{}+ A_{Vj,De} H_{De} + A_{Vj,Ht_{F}} H_{Ht_{F}} + A_{Vj,Hs_{u}} u_{Hs_{F}}, \\& 0 = A_{Lj,Pi} H_{Pi} + A_{Lj,Pu} H_{Pu} \\& \hphantom{0 =}{} + A_{Lj,De} H_{De} + A_{Lj,Ht_{F}} H_{Ht_{F}} + A_{Lj,Hs_{u}} u_{Hs_{F}}, \\& H_{Pi} = B_{Jc}(q_{Pi}) T_{Vj} + B_{Jc}(q_{Pi}) T_{Lj} + B_{Jc}(q_{Pi}) T_{Re}, \\& H_{Pu} = B_{Jc}(q_{Pu}) T_{Vj} + B_{Jc}(q_{Pu}) T_{Lj} + B_{Jc}(q_{Pu}) T_{Re}, \\& H_{Ht_{F}} = c_{Ht_{F}} \bigl( A_{Vj,Ht_{F}}^{T} T_{Vj} + A_{Lj,Ht_{F}}^{T} T_{Lj} + A_{Tb_{u},Ht_{F}}^{T}u_{Tb_{F}} \bigr), \\& y_{Vj} = \bigl\vert \bigl(A_{Vj,Hs_{u}}^{T} + A_{Tb_{u},Ht_{F}}A_{Vj,Ht_{F}}^{T}\bigr) \bigr\vert T_{Vj}, \\& y_{Lj} = \bigl\vert \bigl(A_{Lj,Hs_{u}}^{T} + A_{Tb_{u},Ht_{F}}A_{Lj,Ht_{F}}^{T}\bigr) \bigr\vert T_{Lj}, \\& y_{Ht_{F}} = \bigl(A_{Tb_{u},Ht_{F}} + A_{Lj,Hs_{u}}^{T} A_{Lj,Ht_{F}}^{T} + A_{Vj,Hs_{u}}^{T} A_{Vj,Ht_{F}}^{T}\bigr) H_{Ht_{F}} \end{aligned}$$

(3)

for given boundary conditions \(q_{De} = \bar{q}_{De}\), \(H_{De} = \bar{H}_{De}\), \(p_{Re} = \bar{p}_{Re}\) and \(T_{Re} = \bar{T}_{Re}\) and predefined coefficients \(c_{1,Pi}\), \(c_{2,Pi}\), \(c_{3,Pi}\), \(m_{Vj}\), \(c_{p, Vj}\) and \(c_{Ht_{F}}\) as well as provided functions \(f_{Pu}\). The function \(B_{Jc}\) checks for the sign of the mass flow \(q_{Pi}\), cf. [3]. The coupling variables are expressed as the temperatures \(u_{Hs_{F}}\) and \(u_{Tb_{F}}\) and the energy fluxes \(y_{Vj}\), \(y_{Lj}\) and \(y_{Ht_{F}}\).

### Multi-physical model

The multi-physical model is derived by combining (1), (2) and (3) with appropriate coupling conditions. The coupling conditions describe the relation between the inputs and outputs of the individual models. For the model used in Sect. 3 and Sect. 4, the following coupling conditions are used, see e.g. [2].

\left(\begin{array}{c}{u}_{R}\\ {u}_{C}\\ {u}_{B}\\ {u}_{H{s}_{S}}\\ {u}_{T{b}_{S}}\\ {u}_{H{s}_{F}}\\ {u}_{T{b}_{F}}\end{array}\right)=\left(\begin{array}{ccccccc}0& {C}_{R,Sw}& 0& 0& 0& 0& 0\\ 0& {C}_{C,Sw}& 0& 0& 0& 0& 0\\ 0& {C}_{B,Sw}& 0& 0& 0& 0& 0\\ {C}_{H{s}_{S},R}& 0& 0& 0& 0& 0& {C}_{H{s}_{S},H{t}_{F}}\\ 0& 0& 0& 0& {C}_{T{b}_{S},Vj}& 0& 0\\ 0& 0& 0& {C}_{H{s}_{F},Vj}& 0& 0& 0\\ 0& {C}_{T{b}_{F},Sw}& 0& 0& 0& 0& 0\end{array}\right)\left(\begin{array}{c}{y}_{R}\\ {y}_{Sw}\\ {y}_{Lw}\\ {y}_{H{t}_{S}}\\ {y}_{Vj}\\ {y}_{Lj}\\ {y}_{H{t}_{F}}\end{array}\right).

(4)

The connectivity equation (4) represents the electro-thermal coupling of the electrical network and the cooling systems. Combining all subsystems and their connectivity equations (4) yields a DAE:

Find

$$\begin{aligned}& z :=( u_{R}, u_{C}, u_{B}, u_{Hs_{S}}, u_{Tb_{S}}, u_{Hs_{F}}, u_{Tb_{F}}, e_{N}, e_{G}, j_{R}, j_{C}, j_{L}, j_{V}, j_{B}, T_{Sw}, T_{Lw}, H_{Ht_{S}}, \\& \hphantom{z :=}{} p_{Lj}, p_{Vj}, q_{Pi}, q_{Pu}, T_{Vj}, T_{Lj}, H_{Ht_{F}}, H_{Pu}, H_{Pi}, y_{R}, y_{Sw}, y_{Lw}, y_{Ht_{S}}, y_{Vj}, y_{Lj}, y_{Ht_{F}}), \\& \dot{z} :=\frac{dz}{dt} \end{aligned}$$

such that

$$ F(\dot{z}, z, t) = 0. $$

(5)

DAEs resulting from automated modelling software typically obtain a structure with d-index (differentiation index) greater than 1, cf. [3, 4, 10] and hence are not suitable for a direct simulation with standard solvers. In the setup of multiple physical networks, it is not sufficient, that the full DAE (5) can be reduced to a d-index 1. Additionally, each subsystem, to which a solver is applied, has to fulfill d-index 1 conditions as well, cf. [7, 11]. In our applications an automatic index reduction is performed if the electric or the fluid system are detected to be of d-index 2. This is achieved by providing surrogate models according the corresponding literature, cf. [3, 4], which enables additionally a consistent definition of initial values. Similarly, it is possible to describe the coupling of thermal solid systems with gas systems. An approach for coupling fluid, gas and thermal solid systems can be found in [5].